Finding the linear transformation of a matrix

1. The problem statement, all variables and given/known data
Hello again. First of all thanks to anyone who has replied to my previous questions.
Now, the question that troubles me is:

We are given a matrix A2x2 with some random values and we are asked to say if there is a linear map which has A as its map for the standar basis.

A= 2 1
-5 8

2. Relevant equations



3. The attempt at a solution
If A is the matrix of some linear map for the standar basis that means that:

f(1,0)=2(1,0) -5(0,1)
f(0,1)=1(1,0) +8(0,1)

so f(1,0)=(2,-5) and f(0,1)=(1,8)

Although the question only asks to say if such a linear map exists (it obviously does but i dont know how to prove it) I would be glad if someone could instruct me on finding formulas for linear maps when we have their matrix.
 
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So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

[tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
[tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
[tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
[tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

Can you fill in the missing entries in the 2 x 2 matrix?
 
So given a matrix, you are trying to find the linear transformation represented by the matrix. Building on the work you already have, and identifying the transformation by T, we have.

[tex]T\left(\begin{array}{c}x&y\end{array} \right)[/tex]
[tex]=~x~T\left(\begin{array}{c}1&0\end{array} \right)~+~y~T\left(\begin{array}{c}0&1\end{array} \right)[/tex]
[tex]=~x~\left(\begin{array}{c}2&-5\end{array} \right)~+~y~T\left(\begin{array}{c}1&8\end{array} \right)[/tex]
[tex]=~\left[\begin{array}{cc}?&?&?&?\end{array} \right]~\left(\begin{array}{c}x&y\end{array} \right)[/tex]

Can you fill in the missing entries in the 2 x 2 matrix?
Yes I think I got it now. The missing matrix is the matrix i posted on my first post. So T(x,y)=(2x+y,-5x+8y). The way you put it, I see that a matrix and a linear transformation are actually the exact same thing, expressed in a different way? Thank you very much, the work you people do here is incredible.
 

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